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We study the Erdos distance problem over finite Euclidean and non-Euclideanspaces. Our main tools are graphs associated to finite Euclidean andnon-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004,2007). These graphs are shown to be asymptotically Ramanujan graphs. Theadvantage of using these graphs is twofold. First, we can derive new lowerbounds on the Erdos distance problems with explicit constants. Second, we canconstruct many explicit tough Ramsey graphs R(3,k).

Explicit Ramsey graphs and Erdos distance problem over finite Euclidean and non-Euclidean spaces